TY  - JOUR
T1  - On Strongly 𝝅-regular Modules
AU  - Koç, Suat
PY  - 2020
DA  - August
Y2  - 2020
DO  - 10.16984/saufenbilder.696366
JF  - Sakarya University Journal of Science
JO  - SAUJS
PB  - Sakarya University
WT  - DergiPark
SN  - 2147-835X
SP  - 675
EP  - 684
VL  - 24
IS  - 4
LA  - en
AB  - In this article, we introduce the notion of strongly π-regular module which is a generalization of von Neumann regular module in the sense [13]. Let A be a commutative ring with 1≠0 and X a multiplication A-module. X is called a strongly π-regular module if for each x∈X, 〖(Ax)〗^m=cX=c^2 X for some c∈A and m∈N. In addition to give many properties and examples of strongly π-regular modules, we also characterize certain class of modules such as von Neumann regular modules and second modules in terms of this new class of modules. Also, we determine when the localization of any family of submodules at a prime ideal commutes with the intersection of this family.
KW  - von Neumann regular module
KW  - (m n)-closed ideal
KW  - strongly π-regular module
KW  - Krull dimension
KW  - (∗)-property
KW  - localization
CR  - [1] R. Ameri, “On the prime submodules of multiplication modules,” International journal of Mathematics and mathematical Sciences, vol. 27, pp. 1715-1724, 2003.
CR  - [2] D. D. Anderson, S. Chun and J. R. Juett, “Module-theoretic generalization of co mmutative von Neumann regular rings, “ Communications in Algebra, vol. 47, no. 11, pp. 4713-4728, 2019.
CR  - [3] D. F. Anderson and A. Badawi, “ On (m,n)-closed ideals of commutative rings,” Journal of Algebra and Its Applications, vol. 16, no. 01, 1750013 (21 page), 2017.
CR  - [4] M. F. Atiyah and I. G. Macdonald, “Introduction to commutative algebra,” Westview press, 1994.
CR  - [5] A. Barnard, “Multiplication modules,” Journal of Algebra, vol. 71, no. 1, pp. 174-178, 1981.
CR  - [6] Z. Bilgin and K. H. Oral, “Coprimely structured modules,” Palestine J Math, vol. 7, Spec. Issue 1, pp. 161–169, 2018.
CR  - [7] Z. Bilgin, S. Koç and A. Özkirişci, “Strongly prime submodules and strongly 0-dimensional modules,” Algebra and Discrete Mathematics, vol. 28, no. 2, pp. 171-183, 2019.
CR  - [8] J. Brewer and F. Richman, “Subrings of zero-dimensional rings,” In Multiplicative ideal theory in commutative algebra, Springer, Boston, MA, pp. 73-88, 2006.
CR  - [9] V. Camillo and W. K. Nicholson, “Quasi-morphic rings,” Journal of Algebra and Its Applications, vol. 6, no. 05, pp. 789-799, 2007.
CR  - [10] P. V. Danchev, “A generalization of 𝜋-regular rings,” Turkish Journal of Mathematics, vol. 43, no. 2, pp. 702-711, 2019.
CR  - [11] Z. A. El-Bast and P. F.Smith, “Multiplication modules,” Comm. in Algebra, vol. 16, no. 4, pp. 755-779, 1988.
CR  - [12] M. Evans, “On commutative PP rings,” Pacific Journal of mathematics, vol. 41, no. 3, pp. 687-697, 1972.
CR  - [13] C. Jayaram and Ü. Tekir, “von Neumann regular modules,” Communications in Algebra, vol. 46, no. 5, pp. 2205-2217, 2018.
CR  - [14] I. Kaplansky, “Commutative rings,” Allyn and Bacon, 1970.
CR  - [15] T. K. Lee and Y. Zhou, “Reduced Modules,” Rings, Modules, Algebras and Abelian Groups. Lecture Notes in Pure and Appl. Math., Vol. 236. New York: Dekker, pp. 365-377, 2004.
CR  - [16] C. P. Lu, “Prime submodules of modules,” Comment. Math. Univ. Sanct. Pauli, vol. 33, no. 1, pp. 61–69, 1984.
CR  - [17] E. Matlis, “Divisible modules,” Proceedings of the American Mathematical Society, vol. 11, no. 3, pp. 385-391, 1960.
CR  - [18] K. H. Oral, N. A. Özkirişci and Ü. Tekir, “Strongly 0-dimensional modules, “ Canadian Mathematical Bulletin, vol. 57, no. 1, pp. 159-165, 2014.
CR  - [19] P. Ribenboim, “Algebraic Numbers,” New York, NY, USA: Wiley, 1974.
CR  - [20] R. Y. Sharp, “Steps in commutative algebra,” (No. 51). Cambridge university press, 2000.
CR  - [21] P. F. Smith, “Some remark on multiplication modules”, Archiv der Mathematik, vol. 50, no. 3, pp. 223-235, 1988.
CR  - [22] J. Von Neumann, “On regular rings,” Proceedings of the National Academy of Sciences of the United States of America, vol. 22, no. 12, pp. 707-713, 1936.
CR  - [23] S. Yassemi, “The dual notion of prime submodules,” Arch. Math.(Brno), vol. 37, no. 4, pp. 273-278, 2001.
CR  - [24] H. Zhu and N. Ding, “Generalized morphic rings and their applications, “ Communications in Algebra, vol. 35, no. 9, pp. 2820-2837, 2007.
UR  - https://doi.org/10.16984/saufenbilder.696366
L1  - https://dergipark.org.tr/en/download/article-file/1210245
ER  -